Thumbnail · 2016-07-04 · Module 2: Trigonometric Functions 70 Module 3: Exponential and Logarithmic Functions 79 ... Algebra II lead writer/editor Eureka Math/Great Minds From

Thumbnailjpg

Eureka Math Algebra II Study Guide

Wheatley Portfolio

English Grades Kndash5 Second Edition

English Grades 6ndash8 Second Edition


alexandria Plan

United States History Grades Kndash2

World History Grades Kndash2

United States History Grades 3ndash5

World History Grades 3ndash5

eUreKa Math

Eureka Math Grade K Study Guide

Eureka Math Grade 1 Study Guide








Eureka Math Algebra I Study Guide

Eureka Math Geometry Study Guide

Other Books


Cover design by Chris Clary

Cover image Leonardo da Vinci (1452ndash1519) Study of an old man by the water and also a study of water Photo ScalaArt Resource NY

Copyright copy 2016 by Great Minds All rights reserved

Published by Jossey-BassA Wiley BrandOne Montgomery Street Suite 1000 San Francisco CA 94104-4594mdashwwwjosseybasscom

No part of this publication may be reproduced stored in a retrieval system or transmitted in any form or by any means electronic mechanical photocopying recording scanning or otherwise except as permitted under Section 107 or 108 of the 1976 United States Copyright Act without either the prior written permission of the publisher or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center Inc 222 Rosewood Drive Danvers MA 01923 978-750-8400 fax 978-646-8600 or on the Web at wwwcopyrightcom Requests to the publisher for permission should be addressed to the Permissions Department John Wiley amp Sons Inc 111 River Street Hoboken NJ 07030 201-748-6011 fax 201-748-6008 or online at wwwwileycomgopermissions

Limit of LiabilityDisclaimer of Warranty While the publisher and author have used their best efforts in preparing this book they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives or written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with a professional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages including but not limited to special incidental consequential or other damages Readers should be aware that Internet Web sites offered as citations andor sources for further information may have changed or disappeared between the time this was written and when it is read

Jossey-Bass books and products are available through most bookstores To contact Jossey-Bass directly call our Customer Care Department within the US at 800-956-7739 outside the US at 317-572-3986 or fax 317-572-4002

For more information about Eureka Math visit wwweureka-mathorg

Wiley publishes in a variety of print and electronic formats and by print-on-demand Some material included with standard print versions of this book may not be included in e-books or in print-on-demand If this book refers to media such as a CD or DVD that is not included in the version you purchased you may download this material at httpbooksupportwileycom For more information about Wiley products visit wwwwileycom

Library of Congress Cataloging-in-Publication Data has been applied for and is on file with the Library of Congress

ISBN 978-1-118-81220-4 (paper) ISBN 978-1-118-81320-1 (ebk) ISBN 978-1-118-81340-9 (ebk)

Printed in the United States of America

first edition

PB Printing 10 9 8 7 6 5 4 3 2 1

v

Introduction by Lynne Munson viiFrom the Writer by Chris Black ixForeword by Scott Baldridge xiHow to Use This Book xiii

Chapter 1 Introduction to Eureka Math 1Vision and Storyline 1Advantages to a Coherent Curriculum 2

Chapter 2 Major Mathematical Themes in Each Grade Band 5Year-Long Curriculum Maps for Each Grade Band 5Math Content Development for Grades 9ndash12 A Story of Functions 5How A Story of Functions Aligns with the Instructional Shifts 11How A Story of Functions Aligns with the Standards for Mathematical Practice 14

Chapter 3 Course Content Review 19Rationale for Module Sequence in Algebra II 21

Chapter 4 Curriculum Design 27Approach to Module Structure 27Approach to Lesson Structure 28Approach to Assessment 49

Chapter 5 Approach to Differentiated Instruction 51Scaffolds for English Language Learners 52Scaffolds for Students with Disabilities 53Scaffolds for Students Performing below Grade Level 55Scaffolds for Students Performing above Grade Level 56

Chapter 6 Course Module Summary and Unpacking of Standards 57Module 1 Polynomial Rational and Radical Relationships 58Module 2 Trigonometric Functions 70Module 3 Exponential and Logarithmic Functions 79Module 4 Inferences and Conclusions from Data 95

Contents

vi | Contents

Chapter 7 Terminology 107Algebra I 107Geometry 111Algebra II 113Precalculus and Advanced Topics 117

Notes 129Board of Trustees 133Eureka Math A Story of Functions Contributors 135Index 137

vii

When do you know you really understand something One test is to see if you can explain it to someone elsemdashwell enough that they understand it Eureka Math routinely requires students to ldquoturn and talkrdquo and explain the math they learned to their peers

That is because the goal of Eureka Math (which you may know as the EngageNY math modules) is to produce students who are not merely literate but fluent in mathematics By fluent we mean not just knowing what process to use when solving a problem but understanding why that process works

Herersquos an example A student who is fluent in mathematics can do far more than just name recite and apply the Pythagorean theorem to problems She can explain why a2 + b2 = c2 is true She not only knows that the theorem can be used to find the length of a right trianglersquos hypotenuse but also can apply it more broadlymdashsuch as to find the distance between any two points in the coordinate plane for example She also can see the theorem as the glue joining seemingly disparate ideas including equations of circles trigonometry and vectors

By contrast the student who has merely memorized the Pythagorean theorem does not know why it works and can do little more than just solve right triangle problems by rote The theorem is an abstractionmdashnot a piece of knowledge but just a process to use in the limited ways that she has been directed For her studying mathematics is a chore a mere memorizing of disconnected processes

Eureka Math provides much more It offers students math knowledge that will serve them well beyond any test This fundamental knowledge not only makes wise citizens and competent consumers but also gives birth to budding physicists and engineers Knowing math deeply opens vistas of opportunity

Students become fluent in mathmdashas they do in any other subjectmdashby following a course of study that builds their knowledge of the subject logically and thoroughly In Eureka Math concepts flow logically from PreKindergarten through high school The ldquochaptersrdquo in the story of mathematics are A Story of Units for the elementary grades followed by A Story of Ratios in middle school and A Story of Functions in high school

This sequencing is joined with a mix of new and old methods of instruction that are proven to work For example we utilize an exercise called a ldquosprintrdquo to develop studentsrsquo fluency with standard algorithms (routines for adding subtracting multiplying and dividing whole numbers and fractions) We employ many familiar models and tools such as the number line and tape diagrams (aka bar models) A newer model highlighted in the curriculum is the number bond (illustrated on the following page) which clearly shows how numbers are composed of other numbers

Introduction

viii | IntroductIon

Eureka Math is designed to help accommodate different types of classrooms and to serve as a resource for educators who make decisions based on the needs of students The ldquovignettesrdquo of teacher-student interactions included in the curriculum are not scripts but exemplars illustrating methods of instruction recommended by the teachers who have crafted our curriculum

Eureka Math has been adopted by districts from East Meadows New York to Lafayette Louisiana to Chula Vista California At Eureka Math we are excited to have created the most transparent math curriculum in historymdashevery lesson all classwork and every problem is available online

Many of us have less than joyful memories of learning mathematics lots of memorization lots of rules to follow without understanding and problems that didnrsquot make any sense What if a curriculum came along that gave children a chance to avoid that math anxiety and replaced it with authentic understanding excitement and curiosity Like a New York educator attending one of our trainings said ldquoWhy didnrsquot I learn mathematics this way when I was a kid It is so much easier than the way I learned itrdquo

Eureka

Lynne MunsonWashington DC

ix

To My Fellow Teachers

I was hooked into mathematics by the intricacy and beauty of high school geometry I distinctly remember being home sick from school and constructing a new-to-me proof of the Pythagorean theorem for the sheer joy of solving the puzzle My interest in mathematics grew through subsequent courses in algebra trigonometry and calculus although I didnrsquot fully grasp the connections among these fields of mathematics until I began to teach

Classroom teachers and mathematicians from across the country have come together to develop the Eureka Math curriculum to bring these connections to life For example in Algebra II students will see that we can use the analytic concept of the average rate of change to lead to the geometric formulas for the area of a circle and the volume of a sphere They will see that we can apply geometric concepts of congruence and similarity to parabolas in the coordinate plane and that the study of motion around a circlemdashthe simplest geometric objectmdashleads to an entire field of mathematics that encompasses and expands on the triangle trigonometry studied in Geometry

This course that we share with you is full of deep and beautiful mathematics carefully sequenced to coherently build on the foundations of Algebra I and Geometry previously constructed We have developed and refined the modules with great care however we envision that you will adapt these materials to meet the needs of your particular students collaborating with your colleagues to tinker with the lessons and make them your own

Speaking for the teachers and mathematicians who outlined wrote critiqued rewrote edited and revised these lessons we thank you for the careful deliberation you put into your preparation every day in order to promote student success We consider you to be a crucial part of our team and we look forward to your input on how we can continue to improve the Algebra II curriculum

Chris BlackSeattle WA

Algebra II lead writereditorEureka MathGreat Minds

From the Writer

xi

Foreword

Telling The STory of MaTh

Each module in Eureka Math builds carefully and precisely on the content learned in the previous modules and years weaving the knowledge learned into a coherent whole This produces an effect similar to reading a good novel The storyline even after weeks of not reading is easy to pick up again because the novel pulls the reader back into the plot immediatelymdashthe need to review is minimal because the plot brings out and adds to what has already happened This cumulative aspect of the plot along with its themes character development and composition are all part of the carefully thought-out design of the Eureka Math curriculum

So what is the storyline One can get a sense of how the story evolves by studying the major themes of A Story of Units A Story of Ratios and A Story of Functions

A Story of Units investigates how concepts including place value algorithms fractions measurements area and so on can all be understood by relating and manipulating types of units (eg inches square meters tens fifths) For example quantities expressed in the same units can be added 3 apples plus 4 apples equals 7 apples Likewise 3 fifths plus 4 fifths is 7 fifths Whole number multiplication as in ldquo3 fives = 15 onesrdquo is merely another form of converting between different units as when we state that ldquo1 foot = 12 inchesrdquo These similari-ties between concepts drive the day-to-day theme throughout the PreKndash5 curriculum each type of unit (or building block) is handled the same way through the common features that all units share Understanding the commonalities and like traits of these building blocks makes it much easier to sharply contrast the differences In other words the consistency of manipula-tion of different units helps students see the connection in topics No longer is every new topic separate from the previous topics studied

A Story of Ratios moves students beyond problems that involve one-time calculations using one or two specific measurements to thinking about proportional relationships that hold for a whole range of measurements The proportional relationships theme shows up every day during middle school as students work with ratios rates percentages probability similarity and linear functions A Story of Ratios provides the transition years between students thinking of a specific triangle with side lengths 3 cm 4 cm and 5 cm in elementary school to a broader view in high school for studying the set of all triangles with side lengths in a 345 ratio (eg 6810 91215)

A Story of Functions generalizes linear relationships learned in middle school to polynomial rational trigonometric exponential and logarithmic functions in high school Students study the properties of these functions and their graphs and model with them to move explicitly from real-world scenarios to mathematical representations The algebra learned in middle school is applied in rewriting functions in different forms and solving equations derived from one or more functions The theme drives students to finish high

xii | ForeWord

school knowing not only how to manipulate the major functions used in college but also how to be fully capable of modeling real-life data with an appropriate function in order to make predictions and answer questions

The many ldquolittle eurekasrdquo infused in the storyline of Eureka Math help students learn how to wield the true power of mathematics in their daily lives Experiencing these ldquoaha momentsrdquo also convinces students that the mathematics that drives innovation and advancement in our society is within their reach

Scott BaldridgeLead writer and lead mathematician Eureka Math

Loretta Cox Stuckey and Dr James G Traynham Distinguished Professor of Mathematics Louisiana State University

Co-director Gordon A Cain Center for Science Technology Engineering and Mathematical Literacy

xiii

As a self-study resource these Eureka Math Study Guides are beneficial for teachers in a variety of situations They introduce teachers who are brand new to either the classroom or the Eureka Math curriculum not only to Eureka Math but also to the content of the grade level in a way they will find manageable and useful Teachers already familiar with the curriculum will also find this resource valuable as it allows a meaningful study of the grade-level content in a way that highlights the connections between modules and topics The guidebooks help teachers obtain a firm grasp on what it is that students should master during the year The structure of the book provides a focus on the connections between the standards and the descriptions of mathematical progressions through the grade topic by topic Teachers there-fore develop a multifaceted view of the standards from a thorough analysis of the guide

The Eureka Math Study Guides can also serve as a means to familiarize teachers with adjacent grade levels It is helpful for teachers to know what students learned in the grade level below the one they are currently teaching as well as the one that follows Having an understanding of the mathematical progression across grades enhances the teacherrsquos ability to reach students at their level and ensure they are prepared for the next grade

For teachers schools and districts that have not adopted Eureka Math but are instead creating or adjusting their own curricular frameworks these grade-level study guides offer support in making critical decisions about how to group and sequence the standards for maximal coherence within and across grades Eureka Math serves as a blueprint for these educators in turn the study guides present not only this blueprint but a rationale for the selected organization

The Eureka Math model provides a starting point from which educators can build their own curricular plan if they so choose Unpacking the new standards to determine what skills students should master at each grade level is a necessary exercise to ensure appropriate choices are made during curriculum development The Eureka Math Study Guides include lists of student outcomes mapped to the standards and are key to the unpacking process The overviews of the modules and topics offer narratives rich with detailed descriptions of how to teach specific skills needed at each grade level Users can have confidence in the interpretations of the standards presented as well as the sequencing selected due to the rigorous review process that occurred during the development of the content included in Eureka Math

This Eureka Math Study Guide contains the following

introduction to eureka Math (chapter 1) This introduction consists of two sections ldquoVision and Storylinerdquo and ldquoAdvantages to a Coherent Curriculumrdquo

Major Mathematical Themes in each grade Band (chapter 2) The first section presents year-long curriculum maps for each grade band (with subsections addressing A Story of Units A Story of Ratios and A Story of Functions) It is followed by a detailed examination of math concept development for courses typically taught from Grade 9 to Grade 12 The chapter closes with an in-depth description of how alignment to the Instructional Shifts and the Standards of Mathematical Practice is achieved

How to use this Book

xiv | HoW to use tHIs Book

Course Content review (chapter 3) The purpose and recommended fluencies for the course are presented in this chapter along with a rationale for why topics are grouped and sequenced in the modules as they are The Alignment to the Standards and Placement of Standards in the Modules chart lists the standards that are addressed in each module of the course

Curriculum Design (chapter 4) The approach to modules lessons and assessment in A Story of Functions is detailed in this chapter

approach to Differentiated instruction (chapter 5) This chapter describes the approach to differentiated instruction used in A Story of Functions Special populations such as English language learners students with disabilities students performing above grade level and students performing below grade level are addressed

Course Module Summary and Unpacking of Standards (chapter 6) This chapter presents information from the modules to provide an overview of the content of each and explain the mathematical progression The standards are translated for teachers and a fuller picture is drawn of the teaching and learning that should take place through the school year

Terminology (chapter 7) The terms included in this list were compiled from the New or Recently Introduced Terms portion of the Terminology section of the Module Overviews Terms are listed by course and module number where they are introduced in A Story of Functions and definitions for these terms are provided

eureka Math Algebra II study Guide


Wheatley Portfolio




alexandria Plan





eUreKa Math












Other Books














first edition

PB Printing 10 9 8 7 6 5 4 3 2 1

v








Contents

vi | Contents



vii








Introduction

viii | IntroductIon




Eureka


ix








From the Writer

xi

Foreword







xii | ForeWord






xiii
















Wheatley Portfolio




alexandria Plan





eUreKa Math












Other Books














first edition

PB Printing 10 9 8 7 6 5 4 3 2 1

v








Contents

vi | Contents



vii








Introduction

viii | IntroductIon




Eureka


ix








From the Writer

xi

Foreword







xii | ForeWord






xiii





























first edition

PB Printing 10 9 8 7 6 5 4 3 2 1

v








Contents

vi | Contents



vii








Introduction

viii | IntroductIon




Eureka


ix








From the Writer

xi

Foreword







xii | ForeWord






xiii




























first edition

PB Printing 10 9 8 7 6 5 4 3 2 1

v








Contents

vi | Contents



vii








Introduction

viii | IntroductIon




Eureka


ix








From the Writer

xi

Foreword







xii | ForeWord






xiii
















v








Contents

vi | Contents



vii








Introduction

viii | IntroductIon




Eureka


ix








From the Writer

xi

Foreword







xii | ForeWord






xiii
















vi | Contents



vii








Introduction

viii | IntroductIon




Eureka


ix








From the Writer

xi

Foreword







xii | ForeWord






xiii
















vii








Introduction

viii | IntroductIon




Eureka


ix








From the Writer

xi

Foreword







xii | ForeWord






xiii
















viii | IntroductIon




Eureka


ix








From the Writer

xi

Foreword







xii | ForeWord






xiii
















ix








From the Writer

xi

Foreword







xii | ForeWord






xiii
















xi

Foreword







xii | ForeWord






xiii
















xii | ForeWord






xiii
















xiii
























Documents

Thumbnail · 2016-07-04 · Module 2: Trigonometric Functions 70 Module 3: Exponential and Logarithmic Functions 79 ... Algebra II lead writer/editor Eureka Math/Great Minds From